Question

What is the synthesis equation of the discrete time signal x(n), whose Fourier transform is X(ω)?

(a) \(2π\int_0^2π X(ω) e^jωn dω\)

(b) \(\frac{1}{π} \int_0^{2π} X(ω) e^jωn dω\)

(c) \(\frac{1}{2π} \int_0^{2π} X(ω) e^jωn dω\)

(d) None of the mentioned

The question was posed to me by my school teacher while I was bunking the class.

My doubt stems from Frequency Analysis of Discrete Time Signal topic in chapter Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Right answer is (c) \(\frac{1}{2π} \int_0^{2π} X(ω) e^jωn dω\)

For explanation I would say: We know that the Fourier transform of the discrete time signal x(n) is

X(ω)=\(\sum_{n=-∞}^∞ x(n)e^{-jωn}\)

By calculating the inverse Fourier transform of the above equation, we get

x(n)=\(\frac{1}{2π} \int_0^{2π} X(ω) e^{jωn} dω\)

The above equation is known as synthesis equation or inverse transform equation.